Читать книгу Modern Trends in Structural and Solid Mechanics 1 - Группа авторов - Страница 19
1.3.3. Laminate with arbitrary boundary conditions
ОглавлениеWe now present results for a [0/90/0] laminate simply supported on edges y = 0, b, with the other two edges either clamped or traction-free, and loaded on the top surface by the surface tractions listed in equation [1.12]. For the plate with two edges free, the present results have been computed with N1 = N2 = 11 and N3 = 5 in equation [1.9] for a total of 14,157 DOFs. These values could not be increased due to memory limitation of the Dell laptop used for the computational work. We are now converting the code from MATLAB to C++, which will enable us to use larger values of N1, N2 and N3. In Table 1.5, the numerical results for different quantities for the laminates with a/h = 5 and 10 are compared with their values reported by Vel and Batra (1999), who used the Eshelby–Stroh formalism. The maximum difference of 4.2% in the two sets of values of the six quantities suggests that the least-squares method provides a reasonably accurate solution for this problem.
Table 1.2. Normalized results for the sandwich plate of the aspect ratio a/h = 10. Values in the upper (lower) row are from the analytical (least-squares) solution
| β→ | 1 | 10 | 100 | 1000 | 10,000 | |
|---|---|---|---|---|---|---|
| (0.5, 0.5, -0.5h) | -436.824 (-436.820) | -102.422(-102.422) | -32.9858 (-32.9858) | -21.6946 (-21.6942) | -10.5531(-10.5452) | |
| (0.5, 0.5, z) | ||||||
| z = 0.5h | -24.6675(-24.6687) | -45.1109(-45.1110) | -52.1977(-52.1980) | -155.576(-155.574) | -694.701(-694.429) | |
| z = 0.4h+ | -19.4585(-19.4537) | -32.7715(-32.7713) | -11.4111(-11.4113) | 116.821 (116.819) | 676.521(676.846) | |
| z = 0.4h- | -19.4585(-19.4537) | -3.29448(-3.29446) | -0.13298(-0.13300) | 0.09864(0.09864) | 0.05351(0.05349) | |
| z = -0.4h+ | 19.3977(19.3929) | 3.26259(3.26259) | 0.11506(0.11507) | -0.11458(-0.11457) | -0.06898(-0.06911) | |
| z = -0.4h- | 19.3977(19.3929) | 32.6356(32.6355) | 11.6322(11.6324) | -112.483(-112.471) | -631.675(-632.023) | |
| z = -0.5h | 24.5962(24.5973) | 44.8809(44.8810) | 51.4978(51.4980) | 150.743(150.733) | 649.367(650.206) | |
| (0.5, 0.5, z) | ||||||
| z = 0.5h | -15.6259(-15.6264) | -30.7654(-30.7654) | -48.9480(-48.9478) | -123.891(-123.889) | -442.713(-442.514) | |
| z = 0.4h+ | -12.4002(-12.3978) | -23.1401(-23.1399) | -23.8576(-23.8573) | 43.4841(43.4832) | 399.777(399.971) | |
| z = 0.4h- | -12.4002(-12.3978) | -2.47206(-2.47205) | -0.41064(-0.41064) | -0.12230(-0.12230) | -0.08903(-0.08946) | |
| z = -0.4h+ | 12.3427(12.3404) | 2.33393(2.33392) | 0.23404(0.23404) | -0.05935(-0.05934) | -0.09017(-0.09129) | |
| z = -0.4h- | 12.3427(12.3404) | 23.4274(23.4273) | 24.5512(24.5509) | -40.2356(-40.2337) | -371.648(-374.066) | |
| z = -0.5h | 15.5620(15.5625) | 30.9947(30.9947) | 49.0753(49.0751) | 121.504(121.502) | 415.435(417.599) | |
| (x=0, y=0.5) | ||||||
| z = 0.4h+ | 0.99214(0.99210) | 1.78299(1.78301) | 1.64940(1.64946) | 1.24653(1.24664) | 0.61880(0.59365) | |
| z = -0.4h+ | 0.99152(0.99151) | 1.78239(1.78241) | 1.64878(1.64885) | 1.24591(1.24609) | 0.61820(0.72700) |
Table 1.3. Normalized results for the sandwich plate of the aspect ratio a/h = 5. Values in the upper (lower) row are from the analytical (least-squares) solution
| β→ | 1 | 10 | 100 | 1000 | 10,000 | |
| (0.5, 0.5, -0.5h) | -32.8669(-32.8636) | -11.0284(-11.0283) | -5.97592(-5.97592) | -3.72388(-3.72393) | -0.75513(-0.75586) | |
| (0.5, 0.5, z) | ||||||
| z = 0.5h | -6.24479(-6.24849) | -11.5195(-11.5194) | -22.3767(-22.3764) | -111.720(-111.718) | -325.689(-325.756) | |
| z = 0.4h+ | -4.71502(-4.69630) | -5.97928(-5.97819) | 10.0974(10.0971) | 104.342 (104.339) | 323.587(323.515) | |
| z = 0.4h- | -4.71502(-4.69630) | -0.61521(-0.61510) | 0.08244(0.08244) | 0.08837(0.08836) | 0.02295(0.02297) | |
| z = -0.4h+ | 4.63045(4.61255) | 0.58473(0.58471) | -0.08814(-0.08813) | -0.09061(-0.09061) | -0.02446(-0.02461) | |
| z = -0.4h- | 4.63045(4.61255) | 5.85733(5.85718) | -8.65984(-8.65970) | -86.8644(-86.8631) | -182.172(-183.119) | |
| z = -0.5h | 6.11899(6.12250) | 11.0239(11.0236) | 20.1365.(20.1366) | 93.4349(93.4337) | 183.620(184.363) | |
| (0.5, 0.5, z) | ||||||
| z = 0.5h | -4.28086(-4.28249) | -9.36617(-9.36611) | -20.0972(-20.0970) | -74.2391(-74.2373) | -201.621(-201.636) | |
| z = 0.4h+ | -3.31557(-3.30643) | -5.92194(-5.92142) | -0.10678(-0.10682) | 58.6449(58.6434) | 197.676(197.642) | |
| z = 0.4h- | -3.31557(-3.30643) | -0.74984(-0.74979) | -0.17007(-0.17007) | -0.08703(-0.08704) | -0.06604(-0.06579) | |
| z = -0.4h+ | 3.24276(3.23401) | 0.61151(0.61150) | 0.00125(0.00124) | -0.08155(-0.08154) | -0.06794(-0.06842) | |
| z = -0.4h- | 3.24276(3.23401) | 6.20664(6.20655) | 1.52702(1.52704) | -47.3508(-47.3527) | -110.275(-110.197) | |
| z = -0.5h | 4.18246(4.18400) | 9.41935(9.41922) | 19.2533(19.2532) | 63.5372(63.5391) | 114.682.(114.477) | |
| (X=0,Y=0.5) | ||||||
| z = 0.4h+ | 0.49637(0.49615) | 0.83806(0.83804) | 0.71375(0.71377) | 0.47356(0.47363) | 0.12573(0.10374) | |
| z = -0.4h+ | 0.49166(0.49144) | 0.82533(0.82532) | 0.69746(0.69747) | 0.45722(0.45722) | 0.11274(0.12017) |
Table 1.4. Normalized results for the sandwich plate of the aspect ratio a/h = 2. Values in the upper (lower) row are from the analytical (least-squares) solution
| β→ | 1 | 10 | 100 | 1000 | 10,000 | |
| (0.5, 0.5, -0.5h) | -1.55045(-1.55041) | -0.80673(-0.80672) | -0.47167(-0.47167) | -0.07767(-0.07767) | -0.00174(-0.00179) | |
| (0.5, 0.5, z) | ||||||
| z = 0.5h | -1.30381(-1.30409) | -3.68270(-3.68274) | -17.8693(-17.8692) | -63.5162(-63.5153) | -92.9483(-92.9222) | |
| z = 0.4h+ | -0.71151(-0.70956) | 0.70904(0.70917) | 15.5178(15.5175) | 62.5785(62.5774) | 92.7600(92.7827) | |
| z = 0.4h- | -0.71151(-0.70956) | 0.05412(0.05413) | 0.13952(0.13952) | 0.05564(0.05564) | 0.00816(0.00814) | |
| z = -0.4h+ | 0.53608(0.53463) | -0.03570(-0.03569) | -0.06779(-0.06780) | -0.01411(-0.01411) | -0.00082 | |
| z = -0.4h- | 0.53608(0.53463) | -0.34646(-0.34641) | -6.56097(-6.56090) | -11.5108(-11.5113) | -2.59311(-1.9654) | |
| z = -0.5h | 0.90503(0.90518) | 1.92738(1.92735) | 7.39749(7.39739) | 11.6037(11.6042) | 2.58611(2.41811) | |
| (0.5, 0.5, z) | ||||||
| z = 0.5h | -1.12800(-1.12813) | -3.06623(-3.06624) | -11.9076(-11.9075) | -39.5773(-39.5766) | -57.5749(-57.5841) | |
| z = 0.4h+ | -0.73700(-0.73607) | -0.31380(-0.31374) | 8.78405(8.78389) | 38.4871(38.4863) | 57.3827(57.3665) | |
| z = 0.4h- | -0.73700(-0.73607) | -0.18445(-0.18444) | -0.05495(-0.05496) | -0.02475(-0.02477) | -0.00442(-0.00463) | |
| z = -0.4h+ | 0.58971(0.58899) | 0.06560(0.06560) | -0.04952(-0.04952) | -0.03041(-0.03040) | -0.00524(-0.00247) | |
| z = -0.4h- | 0.58971(0.58899) | 0.75187(0.75190) | -2.96035(-2.96026) | -6.72640(-6.72638) | -1.55007(-1.46223) | |
| z = -0.5h | 0.83614(0.83621) | 2.17320(2.17319) | 5.68229(5.68223) | 7.57969(7.57976) | 1.65537(1.42636) | |
| (X=0, Y=0.5) | ||||||
| z = 0.4h+ | 0.22882(0.22878) | 0.36501(0.36502) | 0.30164(0.30166) | 0.11080(0.11086) | 0.01593(0.03117) | |
| z = -0.4h+ | 0.17656(0.17652) | 0.23854(0.23853) | 0.16531(0.16530) | 0.03726(0.03723) | 0.00302(0.00056) |
Table 1.5. Normalized results for the [0/90/0] laminated plate with C–C–SS–SS and F–F–SS–SS boundary conditions
| a/h | Variable | Clamped–Clamped | Free-Free | ||
|---|---|---|---|---|---|
| Vel and Batra | Present | Vel and Batra | Present | ||
| 5 | (a/2, b/2, h/2) | 1.1800 | 1.1771 | 1.5250 | 1.4649 |
| (a/2, b/2, 0) | -4.2350 | -4.2757 | -6.9870 | -7.2402 | |
| (a/2, b/2, h) | 4.5040 | 4.5437 | 7.1800 | 7.4300 | |
| (a/2, b/2, h/3) | -3.7260 | -3.7185 | -4.784 | -4.582 | |
| (a/2, b/2, 2h/3) | 3.5760 | 3.5652 | 4.639 | 4.4364 | |
| (a/2, 0, h/2) | 1.4700 | 1.4711 | 1.9110 | 1.8494 | |
| 10 | (a/2, b/2, h/2) | 0.4460 | 0.4457 | 0.7530 | 0.72872 |
| (a/2, b/2, 0) | -3.0000 | -2.9746 | -5.8980 | -5.983 | |
| (a/2, b/2, h) | 3.0320 | 3.0066 | 5.9060 | 5.9901 | |
| (a/2, b/2, h/3) | -1.7130 | -1.7113 | -2.882 | -2.7854 | |
| (a/2, b/2, 2h/3) | 1.6740 | 1.672 | 2.845 | 2.748 | |
| (a/2, 0, h/2) | 0.7220 | 0.7243 | 1.2280 | 1.1973 |
